Question
Graph the solutions for the inequality: −3x+1≤−47 (2 points) Responses Question 10 9. A student claims that graph below represents the solutions to the inequality: −4<x What was the student's mistake? (1 point) Responses The student did x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4 The student did x is less than -4, when the variable is on the other side; -4 is less than x so x is greater than -4 The student did not make a mistake; this is the correct graph of the inequality The student did not make a mistake; this is the correct graph of the inequality The student should have multiplied by a negative and switched the direction of the arrow on the graph to go right instead of left The student should have multiplied by a negative and switched the direction of the arrow on the graph to go right instead of left The student should have filled in the point at -4 to show the solution x could be equal to -4
Answers
GPT-4o mini
To solve the first inequality \(-3x + 1 \leq -47\), we can follow these steps:
1. **Subtract 1 from both sides:**
\[
-3x \leq -47 - 1
\]
\[
-3x \leq -48
\]
2. **Divide both sides by -3.** Remember, when we divide by a negative number, we flip the inequality sign:
\[
x \geq \frac{-48}{-3}
\]
\[
x \geq 16
\]
So the solutions for the inequality \( -3x + 1 \leq -47 \) is \( x \geq 16 \).
To graph this solution on a number line:
- Draw a number line.
- Put a closed circle (or dot) at 16 to signify that 16 is included in the solution (since it's \( \geq\)).
- Shade the line to the right of 16 to show all numbers greater than 16 are also solutions.
---
Regarding the second part of your question about the student's mistake regarding the inequality \( -4 < x \):
The correct interpretation of \( -4 < x \) is that \( x\) is greater than \(-4\). The correct graph for this inequality should have an open circle at -4 (indicating that -4 is not included) and should extend to the right.
If the student's graph shows \( x < -4\) instead, the mistake likely comes from misunderstanding the direction of the inequality. Therefore, the correct response would be:
**The student did \(x\) is less than \(-4\), when the variable is on the other side; \(-4\) is less than \(x\) so \(x\) is greater than \(-4.**
1. **Subtract 1 from both sides:**
\[
-3x \leq -47 - 1
\]
\[
-3x \leq -48
\]
2. **Divide both sides by -3.** Remember, when we divide by a negative number, we flip the inequality sign:
\[
x \geq \frac{-48}{-3}
\]
\[
x \geq 16
\]
So the solutions for the inequality \( -3x + 1 \leq -47 \) is \( x \geq 16 \).
To graph this solution on a number line:
- Draw a number line.
- Put a closed circle (or dot) at 16 to signify that 16 is included in the solution (since it's \( \geq\)).
- Shade the line to the right of 16 to show all numbers greater than 16 are also solutions.
---
Regarding the second part of your question about the student's mistake regarding the inequality \( -4 < x \):
The correct interpretation of \( -4 < x \) is that \( x\) is greater than \(-4\). The correct graph for this inequality should have an open circle at -4 (indicating that -4 is not included) and should extend to the right.
If the student's graph shows \( x < -4\) instead, the mistake likely comes from misunderstanding the direction of the inequality. Therefore, the correct response would be:
**The student did \(x\) is less than \(-4\), when the variable is on the other side; \(-4\) is less than \(x\) so \(x\) is greater than \(-4.**